Designs 2002: Further Computational and Constructive Design Theory

371 Pages·2003·14.97 MB·English

by  R. Julian, R. Abel, F. E. Bennett (auth.), W. D. Wallis (eds.)

DESIGNS 2002
Mathematics and Its Applications
Managing Editor:
M. Hazewinkel
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 563
DESIGNS 2002
Further Computational and
Constructive Design Theory
Edited by
W. D. WALLIS
Southern Illinois University
....
"
Kluwer Academic Publishers
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Library of Congress Cataloging-in-Publication
Wallis, Waiter D.
Designs 2002: Further Computational and Constructive Design Theory
ISBN-13:978-1-4613-7958-4 e-ISBN-13:978-1-4613-0245-2
DOl: 10.1007/978-1-4613-0245-2
Copyright 0 2003 by Kluwer Academic Publishers
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Printed on acid-free paper.
Contents
Preface xi
1
1
The Existence of 2-S0LSSOMs
R. Julian R. Abel and F. E. Bennett
1 Introduction 1
2 Direct Constructions 4
3 Recursive Constructions 8
2
23
Conjugate Orthogonal Diagonal Latin Squares with Missing Subsquares
Frank E. Bennett, Beiliang Du and Hantao Zhang
1 Introduction 24
2 Preliminaries 27
3 Direct Construction 30
4 Recursive Construction 33
5 A General Bound 37
6 The Case 2 :S n :S 6 41
vi DESIGNS 2002
3
47
Combinatorial Trades: a Survey of Recent Results
Elizabeth J. Billington
1 Introduction and Preliminaries 47
2 Trades in designs 48
3 Trades in latin squares 53
4 Trade uses 54
5 Latin representations and trades used in tripartite decomposi-
tions 56
6 f-L-way trades 58
7 Graphical trades 60
8 Future directions 62
4
69
Two-stage Generalized Simulated Annealing for the Construction
of Change-Over Designs
Yuk W. Cheng, Deborah J. Street, William H. Wilson
1 Introduction 69
2 Notation and Definitions 71
3 A Two Stage Generalized Simulated Annealing Algorithm 73
4 Performance of the Two-Stage GSA 75
5
81
New Lower Bounds on the Maximum Number of Mutually Orthog
onal Steiner Triple Systems
J. H. Din'itz and P. Dukes
1 Introduction 81
2 Starters over Finite Fields 83
3 Automorphisms and Hill-climbing 92
4 Conclusion 99
Contents vii
6
103
On minimal defining sets in AG(d, 3)
Diane Donovan, Abdallah Khodkar and Anne Penfold Street
1 Background 104
2 Introduction 105
3 Further Preliminaries 116
4 Proof of the Main Theorem 121
5 Sets of good hyperplanes in AG(d, 3) 124
6 Related critical sets 127
7 Appendix 129
7
133
Hadamard Matrices, Orthogonal designs and Construction Algorithms
Stelios Georgiou, Christos Koukouvinos and Jennifer Seberry
1 Algorithms for constructing Hadamard matrices 134
2 On inequivalent Hadamard matrices 163
3 Algorithms for constructing orthogonal designs 176
4 Short amicable sets and Kharaghani type orthogonal designs 193
8
207
Constructing a Class of Designs with Proportional Balance
Ken Gray and Anne Penfold Street
1 Proportionally balanced designs 208
2 The method of construction 211
3 Feasible parameters for Q = 2, 3 and 4 214
4 Constructions motivated by Construction 5 218
viii DESIGNS 2002
9
227
Constructions Using Balanced n-ary Designs
Malcolm Greig
1 Introduction 227
2 Balanced n-ary Designs 229
3 Our Construction 230
4 Some Examples 232
5 Some Non-existence Results 235
6 An Improved Construction 238
7 Quadratic Residues 240
8 Another Improvement 243
9 An Application 246
10 More Difference Families 247
10
255
Sets of Steiner 'friple Systems of Order 9 Revisited
T. S. Griggs and A. Rosa
1 Introd uction 255
2 Large sets 257
3 Results 260
4 Uniformly intersecting sets 263
11
277
Solving Isomorphism Problems for t-Designs
Reinhard Laue
1 Introduction 277
2 Basics 279
3 Moebius Inversions 282
4 Extensions of designs 293
Contents ix
12
301
Finding Double Youden Rectangles
N. C. K. Phillips and D. A. Preece
1 Introduction 301
2 Definitions and Literature 302
3 Examples 302
4 Searching for a 13-cyclic 13 X 40 DYR 306
5 Isomorphism 309
6 Further Properties of our new DYRs 313
7 Check for Balance 314
13
317
Kirkman Triple Systems and their Generalizations: A Survey
Rolf S. Rees, W. D. Wallis
1 Introduction 318
2 Constructions for Kirkman Trirle Systems and Nearly Kirk-
man Triple Systems for all admIssible orders 334
3 Early Generalizations 342
4 Resolvable Packings and Coverings of v points where v =t 0
(mod 3) 348
5 Other Generalizations 359
6 Conclusion and Acknowledgements 363